This entry is about the formal dual to tensoring in the generality of category theory. For the different concept of cotensor product of comodules see there.
In a closed symmetric monoidal category $V$ the internal hom $[-,-] : V^{op} \times V \to V$ satisfies the natural isomorphism
for all objects $v_i \in V$ (prop.). If we regard $V$ as a $V$-enriched category we write $V(v_1,v_2) := [v_1,v_2]$ and this reads
If we now pass more generally to any $V$-enriched category $C$ then we still have the enriched hom object functor $C(-,-) : C^{op} \times C \to V$. One says that $C$ is powered over $V$ if it is in addition equipped also with a mixed operation $\pitchfork : V^{op} \times C \to C$ such that $\pitchfork(v,c)$ behaves as if it were a hom of the object $v \in V$ into the object $c \in C$ in that it satisfies the natural isomorphism
Let $V$ be a closed monoidal category. In a $V$-enriched category $C$, the power of an object $y\in C$ by an object $v\in V$ is an object $\pitchfork(v,y) \in C$ with a natural isomorphism
where $C(-,-)$ is the $V$-valued hom of $C$ and $V(-,-)$ is the internal hom of $V$.
We say that $C$ is powered or cotensored over $V$ if all such power objects exist.
Powers are frequently called cotensors and a $V$-category having all powers is called cotensored, while the word “power” is reserved for the case $V=$ Set. However, there seems to be no good reason for making this distinction. Moreover, the word “tensor” is fairly overused, and unfortunate since a tensor (= a copower) is a colimit, while a cotensor (= power) is a limit.
$V$ itself is always powered over itself, with $\pitchfork(v_1,v_2) := [v_1,v_2]$.
Every locally small category $C$ ($V = (Set,\times)$ ) with all products is powered over Set: the powering operation
of an object $c$ by a set $S$ forms the $|S|$-fold cartesian product of $c$ with itself, where $|S|$ is the cardinality of $S$.
The defining natural isomorphism
is effectively the definition of the product (see limit).
Max Kelly, section 3.7 of Basic concepts of enriched category theory (tac ,pdf)
Francis Borceux, Vol 2, Section 6.5 of Handbook of Categorical Algebra, Cambridge University Press (1994)
Last revised on June 6, 2018 at 15:26:45. See the history of this page for a list of all contributions to it.